Journal of Approximation Theory最新文献

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Extension of the best polynomial operator in generalized Orlicz Spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-26 DOI: 10.1016/j.jat.2025.106174

Sonia Acinas , Sergio Favier , Rosa Lorenzo

In this paper, we consider the best multivalued polynomial approximation operator for functions in an Orlicz Space

L^{φ} (Ω)

. We obtain its characterization involving

ψ^{-}

and

ψ^{+}

, which are the left and right derivative functions of

φ

. And then, we extend the operator to

L^{ψ^{+}} (Ω)

. We also get pointwise convergence of this extension, where the Calderón–Zygmund class

t_{m}^{p} (x)

adapted to

L^{ψ^{+}} (Ω)

plays an important role.

引用次数: 0

Rearrangement-invariant norm inequalities for convolution operators

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-26 DOI: 10.1016/j.jat.2025.106173

Ron Kerman , S. Spektor

<div><div>Let <math><mrow><mi>k</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math>, where, as usual, <math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math> denotes the class of Lebesgue-integrable functions on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> and <math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math> denotes the class of functions on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math> that are Lebesgue-measurable and bounded almost everywhere. Given <math><mrow><mi>f</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math>, set <math><mrow><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mi>k</mi><mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mspace></mspace><mi>d</mi><mi>y</mi><mo>,</mo><mi>x</mi><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>.</mo></mrow></math>We study inequalities of the form <math><mrow><mi>ρ</mi><mrow><mo>(</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub><mi>f</mi><mo>)</mo></mrow><mo>≤</mo><mi>C</mi><mi>σ</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>,</mo></mrow></math>in which <math><mrow><mi>C</mi><mo>></mo><mn>0</mn></mrow></math> is independent of <math><mrow><mi>f</mi><mo>∈</mo><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math>. The functionals <math><mi>ρ</mi></math> and <math><mi>σ</mi></math> are so-called rearrangement-invariant (r.i.) norms on <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mo>+</mo></mrow></msub><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></m

引用次数: 0

Asymptotics of Bergman polynomials for domains with reflection-invariant corners

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-21 DOI: 10.1016/j.jat.2025.106172

Erwin Miña-Díaz , Aron Wennman

We study the asymptotic behavior of the Bergman orthogonal polynomials

{(p_{n})}_{n = 0}^{\infty}

for a class of bounded simply connected domains

D

. The class is defined by the requirement that conformal maps

φ

D

onto the unit disk extend analytically across the boundary

L

D

, and that

φ^{'}

has a finite number of zeros

z_{1}, \dots, z_{q}

L

. The boundary

L

is then piecewise analytic with corners at the zeros of

φ^{'}

. A result of Stylianopoulos implies that a Carleman-type strong asymptotic formula for

p_{n}

holds on the exterior domain

ℂ ∖ \bar{D}

. We prove that the same formula remains valid across

L ∖ {z_{1}, \dots, z_{q}}

and on a maximal open subset of

D

. As a consequence, the only boundary points that attract zeros of

p_{n}

are the corners. This is in stark contrast to the case when

φ

fails to admit an analytic extension past

L

, since when this happens the zero counting measure of

p_{n}

is known to approach the equilibrium measure for

L

along suitable subsequences.

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引用次数: 0

Exact asymptotic order for generalised adaptive approximations

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-21 DOI: 10.1016/j.jat.2025.106171

Marc Kesseböhmer, Aljoscha Niemann

In this note, we present an abstract approach to study asymptotic orders for adaptive approximations with respect to a monotone set function

J

defined on dyadic cubes. We determine the exact upper order in terms of the critical value of the corresponding

J

-partition function, and we are able to provide upper and lower bounds in terms of fractal-geometric quantities. With properly chosen

J

, our new approach has applications in many different areas of mathematics, including the spectral theory of Kreĭn–Feller operators, quantisation dimensions of compactly supported probability measures, and the exact asymptotic order for Kolmogorov, Gel'fand and linear widths for Sobolev embeddings into the Lebesgue space

L_{ν}^{p}

引用次数: 0

Relations between Kondratiev spaces and refined localization Triebel–Lizorkin spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-18 DOI: 10.1016/j.jat.2025.106162

Markus Hansen , Benjamin Scharf, Cornelia Schneider

We investigate the close relation between certain weighted Sobolev spaces (Kondratiev spaces) and refined localization spaces from Triebel (2006), Triebel (2008). In particular, using a characterization for refined localization spaces from Scharf (2014), we considerably improve an embedding from Hansen (2013). This embedding is of special interest in connection with convergence rates for adaptive approximation schemes.

引用次数: 0

A point process on the unit circle with antipodal interactions

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-03-07 DOI: 10.1016/j.jat.2025.106161

Christophe Charlier

<div><div>We introduce the point process</div><div><math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>∏</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo><</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow></msub><msup><mrow><mrow><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msup><mo>|</mo></mrow></mrow><mrow><mi>β</mi></mrow></msup><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>d</mi><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>,</mo><mspace></mspace><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow><mo>,</mo><mspace></mspace><mi>β</mi><mo>></mo><mn>0</mn><mo>,</mo></mrow></math></div><div>where <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is the normalization constant. This point process is attractive: it involves <math><mi>n</mi></math> dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied C<math><mi>β</mi></math>E involves <math><mi>n</mi></math> uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form <math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></math> as <math><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></math>, where <math><mrow><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msup></mrow></math> and <math><mrow><mn>2</mn><mi>π</mi></mrow></math>-periodic. We prove that the leading order fluctuations around the mean are of order <math><mi>n</mi></math> and given by <math><mrow><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mo>−</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mo>−</mo><mi>π</mi></mrow><mrow><mi>π</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow><mfrac><mrow><mi>d</mi><mi>θ</mi></mrow><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mi>n</mi></mrow></math>, where <math><mrow><mi>U</mi><mo>∼</mo><mi>Uniform</mi><mrow><mo>(</mo><mo>−</mo><mi>π</mi><mo>,</mo><mi>π</mi><mo>]</mo></mrow></mrow></math>. We also prove that the subleading fluctuations around the mean are

引用次数: 0

A note on diffusion limits for stochastic gradient descent

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-02-27 DOI: 10.1016/j.jat.2025.106160

Alberto Lanconelli, Christopher S.A. Lauria

In the machine learning literature stochastic gradient descent has recently been widely discussed for its purported implicit regularization properties. Much of the theory, that attempts to clarify the role of noise in stochastic gradient algorithms, has approximated stochastic gradient descent by a stochastic differential equation with Gaussian noise. We provide a rigorous theoretical justification for this practice that showcases how the Gaussianity of the noise arises naturally.

引用次数: 0

Pseudo s-numbers of embeddings of Gaussian weighted Sobolev spaces

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-02-27 DOI: 10.1016/j.jat.2025.106159

Van Kien Nguyen

In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space

W_{p}^{α} (R^{d}, γ)

of mixed smoothness

α \in N

with error measured in the Gaussian-weighted space

L_{q} (R^{d}, γ)

. We obtain the exact asymptotic order of some pseudo

s

-numbers for the cases

1 \leq q < p < \infty

and

p = q = 2

. Additionally, we also obtain an upper bound and a lower bound for some pseudo

s

-numbers of the embedding of

W_{2}^{α} (R^{d}, γ)

into

L_{\infty}^{\sqrt{g}} (R^{d})

. Our result is an extension of that obtained in Dinh Dũng and Van Kien Nguyen (IMA Journal of Numerical Analysis, 2023) for approximation and Kolmogorov numbers.

引用次数: 0

Relative asymptotics of multiple orthogonal polynomials for Nikishin systems of two measures

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-02-18 DOI: 10.1016/j.jat.2025.106158

A. López García , G. López Lagomasino

We study the relative asymptotics of two sequences of multiple orthogonal polynomials corresponding to two Nikishin systems of measures on the real line, the second one of which is obtained from the first one perturbing the generating measures with non-negative integrable functions. Each Nikishin system consists of two measures.

引用次数: 0

The Pearcey integral in the highly oscillatory region II

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory

Pub Date : 2025-02-13 DOI: 10.1016/j.jat.2025.106150

Chelo Ferreira , José L. López , Ester Pérez Sinusía

We consider the Pearcey integral

P (x, y)

for large values of

| x |

and bounded values of

| y |

. The standard saddle point analysis is difficult to apply because the Pearcey integral is highly oscillating in this region. To overcome this problem we use the modified saddle point method introduced in López et al. (2009). A complete asymptotic analysis is possible with this method, and we derive a complete asymptotic expansion of

P (x, y)

for large

| x |

, accompanied by the exact location of the Stokes lines. There are two Stokes lines that divide the complex

x -

plane in two different sectors in which

P (x, y)

behaves differently when

| x |

is large. The asymptotic approximation is the sum of two asymptotic series whose terms are elementary functions of

x

and

y

. Both of them are of Poincaré type; one of them is given in terms of inverse powers of

x

; the other one in terms of inverse powers of

x^{1 / 2}

, and it is multiplied by an exponential factor that behaves differently in the two mentioned sectors. Some numerical experiments illustrate the accuracy of the approximation.

我们考虑的是 Pearcey 积分 P(x,y)，适用于 |x| 的大值和 |y| 的有界值。标准的鞍点分析难以应用，因为皮尔斯积分在这一区域高度振荡。为了克服这个问题，我们采用了 López 等人（2009 年）提出的修正鞍点方法。我们得出了 P(x,y) 在大|x|时的完整渐近展开，并给出了斯托克斯线的精确位置。有两条斯托克斯线将复 x 平面划分为两个不同的扇形区域，当 |x| 较大时，P(x,y) 在这两个扇形区域的表现不同。近似值是两个近似级数之和，其项是 x 和 y 的初等函数。这两个近似级数都是 Poincaré 类型；其中一个用 x 的反幂表示，另一个用 x1/2 的反幂表示，并乘以一个指数因子，在上述两个扇形中表现不同。一些数值实验说明了近似值的准确性。

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